## The Annals of Statistics

### Semiparametrically point-optimal hybrid rank tests for unit roots

#### Abstract

We propose a new class of unit root tests that exploits invariance properties in the Locally Asymptotically Brownian Functional limit experiment associated to the unit root model. The invariance structures naturally suggest tests that are based on the ranks of the increments of the observations, their average and an assumed reference density for the innovations. The tests are semiparametric in the sense that they are valid, that is, have the correct (asymptotic) size, irrespective of the true innovation density. For a correctly specified reference density, our test is point-optimal and nearly efficient. For arbitrary reference densities, we establish a Chernoff–Savage-type result, that is, our test performs as well as commonly used tests under Gaussian innovations but has improved power under other, for example, fat-tailed or skewed, innovation distributions. To avoid nonparametric estimation, we propose a simplified version of our test that exhibits the same asymptotic properties, except for the Chernoff–Savage result that we are only able to demonstrate by means of simulations.

#### Article information

Source
Ann. Statist., Volume 47, Number 5 (2019), 2601-2638.

Dates
Revised: June 2018
First available in Project Euclid: 3 August 2019

https://projecteuclid.org/euclid.aos/1564797858

Digital Object Identifier
doi:10.1214/18-AOS1758

Mathematical Reviews number (MathSciNet)
MR3988767

#### Citation

Zhou, Bo; van den Akker, Ramon; Werker, Bas J. M. Semiparametrically point-optimal hybrid rank tests for unit roots. Ann. Statist. 47 (2019), no. 5, 2601--2638. doi:10.1214/18-AOS1758. https://projecteuclid.org/euclid.aos/1564797858

#### References

• Ahn, S. K., Fotopoulos, S. B. and He, L. (2001). Unit root tests with infinite variance errors. Econometric Rev. 20 461–483.
• Bickel, P. J. (1982). On adaptive estimation. Ann. Statist. 10 647–671.
• Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1998). Efficient and Adaptive Estimation for Semiparametric Models. Springer, New York. Reprint of the 1993 original.
• Brockwell, P. J. and Davis, R. A. (2016). Introduction to Time Series and Forecasting, 3rd ed. Springer Texts in Statistics. Springer, Cham.
• Callegari, F., Cappuccio, N. and Lubian, D. (2003). Asymptotic inference in time series regressions with a unit root and infinite variance errors. J. Statist. Plann. Inference 116 277–303.
• Cassart, D., Hallin, M. and Paindaveine, D. (2010). On the estimation of cross-information quantities in rank-based inference. In Nonparametrics and Robustness in Modern Statistical Inference and Time Series Analysis: A Festschrift in Honor of Professor Jana Jurečková. Inst. Math. Stat. (IMS) Collect. 7 35–45. IMS, Beachwood, OH.
• Chan, N. H. and Wei, C. Z. (1988). Limiting distributions of least squares estimates of unstable autoregressive processes. Ann. Statist. 16 367–401.
• Chernoff, H. and Savage, I. R. (1958). Asymptotic normality and efficiency of certain nonparametric test statistics. Ann. Math. Stat. 29 972–994.
• Choi, I. (2015). Almost All About Unit Roots. Themes in Modern Econometrics. Cambridge Univ. Press, New York.
• Dickey, D. A. and Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. J. Amer. Statist. Assoc. 74 427–431.
• Dickey, D. A. and Fuller, W. A. (1981). Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49 1057–1072.
• Dufour, J.-M. and King, M. L. (1991). Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary or nonstationary $\mathrm{AR}(1)$ errors. J. Econometrics 47 115–143.
• Elliott, G. and Müller, U. K. (2006). Minimizing the impact of the initial condition on testing for unit roots. J. Econometrics 135 285–310.
• Elliott, G., Rothenberg, T. J. and Stock, J. H. (1996). Efficient tests for an autoregressive unit root. Econometrica 64 813–836.
• Hájek, J. and Šidák, Z. (1967). Theory of Rank Tests. Academic Press, New York.
• Hallin, M. and Puri, M. L. (1988). Optimal rank-based procedures for time series analysis: Testing an ARMA model against other ARMA models. Ann. Statist. 16 402–432.
• Hallin, M. and Puri, M. L. (1994). Aligned rank tests for linear models with autocorrelated error terms. J. Multivariate Anal. 50 175–237.
• Hallin, M., van den Akker, R. and Werker, B. J. M. (2011). A class of simple distribution-free rank-based unit root tests. J. Econometrics 163 200–214.
• Hallin, M., van den Akker, R. and Werker, B. J. M. (2015). On quadratic expansions of log-likelihoods and a general asymptotic linearity result. In Mathematical Statistics and Limit Theorems 147–165. Springer, Cham.
• Hasan, M. N. (2001). Rank tests of unit root hypothesis with infinite variance errors. J. Econometrics 104 49–65.
• Jansson, M. (2008). Semiparametric power envelopes for tests of the unit root hypothesis. Econometrica 76 1103–1142.
• Jeganathan, P. (1995). Some aspects of asymptotic theory with applications to time series models. Econometric Theory 11 818–887.
• Jeganathan, P. (1997). On asymptotic inference in linear cointegrated time series systems. Econometric Theory 13 692–745.
• Klaassen, C. A. J. (1987). Consistent estimation of the influence function of locally asymptotically linear estimators. Ann. Statist. 15 1548–1562.
• Kreiss, J.-P. (1987). On adaptive estimation in stationary ARMA processes. Ann. Statist. 15 112–133.
• Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer Series in Statistics. Springer, New York.
• Müller, U. K. (2011). Efficient tests under a weak convergence assumption. Econometrica 79 395–435.
• Müller, U. K. and Elliott, G. (2003). Tests for unit roots and the initial condition. Econometrica 71 1269–1286.
• Müller, U. K. and Watson, M. W. (2008). Testing models of low-frequency variability. Econometrica 76 979–1016.
• Patterson, K. (2011). Unit Root Tests in Time Series. Vol. 1. Palgrave Texts in Econometrics. Palgrave Macmillan, New York.
• Patterson, K. (2012). Unit Root Tests in Time Series. Vol. 2. Palgrave Texts in Econometrics. Palgrave Macmillan, New York.
• Phillips, P. C. B. (1987). Time series regression with a unit root. Econometrica 55 277–301.
• Phillips, P. C. B. and Perron, P. (1988). Testing for a unit root in time series regression. Biometrika 75 335–346.
• Rothenberg, T. J. and Stock, J. H. (1997). Inference in a nearly integrated autoregressive model with nonnormal innovations. J. Econometrics 80 269–286.
• Rudin, W. (1987). Real and Complex Analysis, 3rd ed. McGraw-Hill, New York.
• Saikkonen, P. and Luukkonen, R. (1993). Point optimal tests for testing the order of differencing in ARIMA models. Econometric Theory 9 343–362.
• Schick, A. (1986). On asymptotically efficient estimation in semiparametric models. Ann. Statist. 14 1139–1151.
• van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge.
• White, J. S. (1958). The limiting distribution of the serial correlation coefficient in the explosive case. Ann. Math. Stat. 29 1188–1197.
• Zhou, B., van den Akker, R. and Werker, B. J. M. (2019). Supplement to “Semiparametrically point-optimal hybrid rank tests for unit roots.” DOI:10.1214/18-AOS1758SUPP.

#### Supplemental materials

• Supplement to “Semiparametrically optimal hybrid rank tests for unit roots”. This supplemental file contains technical proofs for propositions and theorems in the main context.